At the time of it's release I imagine it was. The threat was still very really. It was only four or five years after it's release that the USSR collapsed.
While it's not "scary" for me, as I was too young to understand the threat posed by M.A.D, when I first watched it as a teen, I found it horribly bleak and depressing (I still do actually as I have it on DVD and ripped to my NAS, so it's on rotation). Same with quite a few nuclear holocaust films. The other stand out one is "When the wind blows". It may be a cartoon, but it's up there with Threads as a hugely depressing and bleak film. It's kind of like Britain's Grave Of The Fireflies. Which is funny since just a few years before WTWB, the same people made The Snowman, which is a feel good institution.